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While I'm in Windows Explorer renaming function, hit the "Quick Paste Menu" shortcut, Explorer quits renaming.
That's a tricky one.. Exiting renaming function is probably unavoidable windows thing.. but if CHS could know you were in renaming function, it could simulate the F2 keypress to start a rename, but that's tricky.
However, I may have a little trick for you

CHS waits a small bit of time between when you select an item from quickpaste and when it actually tries to paste it (in order to give the target window time to regain focus).
I think you may find that if you hit F2 right after you select an item from the QuickPaste menu, you'll restart the rename mode and the CHS paste will properly occur in the rename.
Unfortunately that won't help you if you are just trying to quick paste into the middle of an existing filename, rather than completely replacing it.
The best solution in that case might be a small third part shell extension file renamer that would bring up a proper window for renaming files that would let you invoke CHS on the rename editbox as in a normal app.